Eigenvalues of Circulant Matrices and a Conjecture of Ryser

Abstract

We prove that there is no circulant Hadamard matrix H with first row [h1,…,hn] of order n > 4, under some linear conditions on the hi’s. All these conditions hold in the known case n = 4, so that our results can be thought as characterizations of properties that only hold when n = 4. Our first conditions imply that some eigenvalue λ of H is a sum of √ -- n terms h jωj, where ω is a primitive n-th root of 1. The same conclusion holds also if some complex arithmetic means associated to λ are algebraic integers (second conditions). Moreover, our third conditions, related to the recent notion of robust Hadamard matrices, implies also the nonexistence of these circulant Hadamard matrices. If some of the conditions fail, it appears (to us) very difficult to be able to prove the result.